The forming window 4f handbook jackhumechanicsofsheetmetalformingsecond

Fracture Fracture Tearing Wrinkling Compression 1 −2 e 1 e 2 Forming window Figure 5.24 The forming window for plane stress forming of sheet. Note: The approximate solution for Equation 5.12 is arrived at as follows. The equation can be written as 1 − n − ε u n exp n − ε u n = 1 + dA A 1 n As indicated n − ε u and dA A are small compared with unity and the above equation may be approximated as 1 − n − ε u n 1 + n − ε u n = 1 − n − ε u n 2 ≈ 1 + dA A 1 n This gives Equation 5.13.

5.7 Exercises

Ex. 5.1 Cold-rolled steel obeys the law σ f = Kε + ε n . a Determine the strain at which the maximum load is reached in a uniform tensile strip. b What happens when ε n ? [Ans: a n − ε ; b ε ∗ 1 = 0 ] Ex. 5.2 Figure 5.25 shows a 100 mm length of a tensile test-piece in which 10 mm has a width of 12.4 mm and the remainder 12.5 mm. The thickness is uniform at the start, t = 1.2 mm. The material obeys an effective stress strain law σ = 750ε 0.22 MPa. Assuming that each length deforms in uniaxial tension, determine the maximum load and the final 80 Mechanics of Sheet Metal Forming 90 10 12.4 12.5 1.2 Figure 5.25 Dimensions of test-piece for Exercise 5.2. length of a 20 mm gauge length in the wider section and the maximum strain in this section. [Ans: P max = 6 .42 KN, ε 1 A = 0 .17 , l = 23 .7 mm] Ex. 5.3 A method is proposed for measuring the strain-hardening index in sheet as defined in Section 1.1.3. A test-piece is used that has two parallel reduced lengths, one is 10.0 mm width and the other 9.8 mm width. In the wider section a gauge length of 50 mm is marked. The strip is pulled to failure and the gauge length measured to determine the true strain ε a . Obtain a diagram relating the true strain ε a to the strain-hardening index n for the range 0.05 ε a 0.2. Ex. 5.4 A strip subjected to tension consists of two regions of equal length l, one of cross-sectional area A a the other A b . The material is perfectly plastic but is rate sensitive so that the effective stress strain rate law is σ = B ˙ε eff. m . If the extension rate of the combined strip is v, determine the strain rate in each section, ˙ε 1 and ˙ε 2 . Ans : v l 1 + A a A b 1 m ; v l 1 + A b A a 1 m Ex. 5.5 An element of material has an imperfection characterized by f = 0.995 as shown in Figure 5.12. It is deformed in equal biaxial tension, σ 1a = σ 2a . The material obeys an effective stress strain law σ = 600 0.004 + ε 0.2 MPa. Determine the principal stresses and the stress ratio in the groove when the uniform region starts to deform. [Ans: 199.9, 197.9, 0.990] Load instability and tearing 81